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Let $X$ be a Banach space and $T$ be a bounded linear operator on $X$. We denote by $S\left(T\right)$ the set of all complex $\lambda \in ℂ$ such that $T$ does not have the single-valued extension property at $\lambda$. In this note we prove equality up to $S\left(T\right)$ between the left Drazin spectrum, the upper semi-B-Fredholm spectrum and the semi-essential approximate point spectrum. As applications, we investigate generalized Weyl’s theorem for operator matrices and multiplier operators.

Let $T$ be a Banach space operator. In this paper we characterize $a$-Browder’s theorem for $T$ by the localized single valued extension property. Also, we characterize $a$-Weyl’s theorem under the condition $E^a\left(T\right)={\pi }^a\left(T\right),$ where $E^a\left(T\right)$ is the set of all eigenvalues of $T$ which are isolated in the approximate point spectrum and ${\pi }^a\left(T\right)$ is the set of all left poles of $T.$ Some applications are also given.